Fibonacci Sequence: Overview, Questions, Preparation

The Fibonacci Sequence is a series of positive numbers that follow a definite pattern depending on the recurrence relation. Each number in the series results from the sum of the concurrent and previous numbers.

Formula to Find the Fibonacci Sequence

The formula that represents the Fibonacci Series of numbers is:

F_n = F_n-1 + F_n-2

Properties of the Fibonacci Sequence

The Fibonacci Sequence of numbers has two distinct properties that set them apart from other types of numbers. They are:
Upon adding three consecutive numbers in the series, and then dividing the resultant number by two, you will get the number three as the final resultant.

For example, if you take the numbers 1, 2, and 3, and add them, you get the resultant value, 6. And upon dividing this resulting value by two, you get the number ‘3’ as the final resultant value.

If you take four consecutive numbers from the Fibonacci Series (except ‘zero’), multiply the first number with the fourth number and second number with the third number, and then subtract the two resultant numbers, the result value that you get is always ‘one.’

For example, let’s take the numbers 2, 3, 5, and 8. Now by multiplying two with eight, you get 16. And by multiplying three with five, you get 15. When you subtract 15 from 16, you get the resultant value, ‘one.’

The Fibonacci Sequence is one topic for class 11 that holds immense value for the exams and pursuing higher studies, especially Engineering. The formula to find the Fibonacci Sequence and its applications and properties are the two key concepts you learn in the chapter. Students should expect at least one question of 2-3 marks from Fibonacci Sequence. 

1. Given the Fibonacci Sequence, find the sum of the first four terms in the series. 

Solution:

We know that the first four terms of the Fibonacci Sequence are 0, 1, 1, and 2. Now, if we add all the terms, we get 0 + 1 + 1 + 2 = 4.

So, the sum of the first four terms in the Fibonacci Sequence is 4.

2. If the Fibonacci Sequence is denoted by a_n = a_n-1 + a_n-2, n > 2, find a_n+1/a_n for n = 1, 2, 3, 4, 5.

Solution:

As per the question, by substituting the values, we get a₂/a₁, a₃/a₂, a₄/a₃, a₅/a₄, and a₆/a₅.

Now, a₂/a₁ becomes (1+0)/(0-1) = 1 / -1 = -1.

Similarly, a₃/a₂ = 1/1 = 1.

a₄/a₃ = 2/1 = 2.

a₅/a₄ = 3/2 = 1.5.

And finally, a₆/a₅ = 5/3.
3. What are the first six numbers in the Fibonacci Sequence? 

Solution:

We know that the formula for the Fibonacci sequence is:

F_n = F_n-1 + F_n-2.

So, going by this formula, the first six terms in the sequence are 0, 1, 1, 2, 3, 5.

Q: What is the Fibonacci Sequence of Numbers?

Q: What is the Formula to find the Fibonacci Sequence?

Q: What are the Properties of the Fibonacci Sequence of Numbers?

Q: What are the first five numbers of the Fibonacci Sequence? 

Q: What are the Applications of the Fibonacci Sequence?